Structural shape optimization of three dimensional finite element models

TR Number



Journal Title

Journal ISSN

Volume Title


Virginia Tech


The thesis presents a three dimensional shape optimization program which analyzes models made up of linear isoparametric elements. The goal of the program is to achieve a near uniform model stress state and thereby to minimize material volume.

The algorithm is iterative, and performs two analyses per iteration. The first analysis is a static stress analysis of the model for one or more load cases. Based on results from the static analysis, an expansion analysis is performed. Model elements are expanded or contracted based on whether they are stressed higher or lower than a reference stress. The shape changing is done by creating an expansion load vector using the differences between the calculated element stresses and the reference stress. Expansion displacements are solved for, and instead of using them to calculate stresses, the displacements are added to the nodal coordinates to reshape the structure. This process continues until a user defined convergence tolerance is met.

Four programs were used for the analysis process. Models were created using a finite element modeling program called I-IDEAS or CAIEDS. The I-IDEAS output files were converted to input files for the optimizer by a conversion program. The model was optimized using the shape optimization process described above. Post- processing was done using a program written with a graphical programming language called graPHIGS.

Models used to test the program were: a cylindrical pressure vessel with nonuniform thickness, a spherical pressure vessel with non-uniform thickness, a torque arm, and a draft sill casting o a railroad hopper car. Results were compared to similar studies from selected references.

Both pressure vessels converged to near uniform thicknesses, which compared ell with the reference work. In a two dimensional analysis, the torque arm volume decreased 24 percent, which compared well with published results. A three dimensional analysis showed a volume reduction of l3 percent, but there were convergence problems. Finally, the draft sill casting was reduced in volume by 9 percent from a manually optimized design.